Results of an all-sky high-frequency Einstein@Home search for continuous gravitational waves in LIGO's fifth science run

We present results of a high-frequency all-sky search for continuous gravitational waves from isolated compact objects in LIGO's 5th Science Run (S5) data, using the computing power of the Einstein@Home volunteer computing project. This is the only dedicated continuous gravitational wave search that probes this high frequency range on S5 data. We find no significant candidate signal, so we set 90%-confidence level upper-limits on continuous gravitational wave strain amplitudes. At the lower end of the search frequency range, around 1250 Hz, the most constraining upper-limit is $5.0\times 10^{-24}$, while at the higher end, around 1500 Hz, it is $6.2\times 10^{-24}$. Based on these upper-limits, and assuming a fiducial value of the principal moment of inertia of $10^{38}$kg$\,$m$^2$, we can exclude objects with ellipticities higher than roughly $2.8\times10^{-7}$ within 100 pc of Earth with rotation periods between 1.3 and 1.6 milliseconds.


Introduction
Ground-based gravitational wave (GW) detectors will be able to detect a continuous gravitational wave signal from a spinning deformed compact object provided that it is spinning with a rotational period between roughly 1 and 100 milliseconds, that it is sufficiently close to Earth and it is sufficiently "bumpy". Blind searches for continuous gravitational waves probe the whole sky and broad frequency ranges, looking for this type of objects.
In this paper, we present the results of an all-sky Ein-stein@Home search for continuous, nearly monochromatic, high-frequency gravitational waves in data from LIGO's 5th Science Run (S5). A number of searches have been carried out on LIGO data [9,8,3,2,6,5] targeting lower frequency ranges. The only other search covering frequencies up to 1500 Hz was conducted on S6 data [10] taken at least 3 years apart from the data used here. Our search results are only 33% less sensitive than those of Abbot et al [10], even though the S5 data is less sensitive than the S6 data by more than a factor of 2. The search method presented here anticipates the procedure that will be used on the advanced detector (aLIGO) data. This search can be considered an extension of the S5 Einstein@Home search [2] although it employs a different search technique: this search uses the Global Correlation Transform (GCT) method to combine results from coherent F -statistic searches [15,16], as opposed to the previous Einstein@Home search [2] that employed the Hough-transform method to perform this combination. In the end, at fixed computing resources, these two search methods are comparable in sensitivity. However, a semi-coherent F -statistic search is more ef-ficient when considering a broad spin-down range, and for the Einstein@Home searches we have decided to adopt it as our "work horse".
We do not find any significant signal(s) among the set of searched waveforms. Thus, we set 90%-confidence upper-limits on continuous gravitational wave strain amplitudes; near the lower end of the search frequency range between 1253.217-1255.217 Hz, the most constraining upper-limit is 5.0 × 10 −24 , while toward the higher end of the search frequency range nearing 1500 Hz, the upper-limit value is roughly 6.2 × 10 −24 . Based on these upper-limits, we can exclude certain combinations of signal frequency, star deformation (ellipticity) and distance values. We show with this search that even with S5 data from the first generation of GW detectors, such constraints do probe interesting regions of source parameter space.

The Data
The LIGO gravitational wave network consists of two detectors, H1 in Hanford (Washington) and L1 in Livingston (Louisiana), separated by a 3000-km baseline. The S5 run lasted roughly two years between GPS time 815155213 sec (Fri, Nov 04, 16:00:00 UTC 2005) and 875145614 sec (Sun, Sep 30, 00:00:00 UTC 2007). This search uses data spanning this observation period, and during this time, H1 and L1 had duty-factors of 78% and 66% respectively [7,4]. The gaps in this data-set are due to environmental or instrumental disturbances, or scheduled maintenance periods.
We follow [2,5], where the calibrated and high-pass filtered data from each detector is partitioned in 30minute chunks and each chunk is Fourier-transformed a avneet.singh@aei.mpg.de b maria.alessandra.papa@aei.mpg.de after the application of a steep Tukey window. The set of Short (time-baseline) Fourier Transforms (abbrev. SFT) that ensues, is the input data for our search.
We further follow [2], where frequency bands known to contain spectral disturbances have been removed from the analysis. In fact, such data has been substituted with fake Gaussian noise at the same level as the neighboring undisturbed data; in table 3, we list these bands.

The Search
The search presented here is similar to the search on S6 data, reported in [9]. Our reference target signal is given by (1)-(4) in [8]; at emission, the signal is nearly monochromatic, typically with a small 'spin-down'. The signal waveform in the detector data is modulated in frequency because of the relative motion between the compact object and the detector; a modulation in amplitude also occurs because of the variation of the sensitivity of the detector with time across the sky.
The most sensitive search technique that one could use is a fully-coherent combination of the detectors' data, matched to the waveform that one is looking for. The (amplitude) sensitivity of such a method increases with the square-root of the time-span of the data used. However, the computational cost to resolve different waveforms increases very rapidly with increasing timespan of the data, and this makes a fully-coherent search over a large frequency range computationally unfeasible when using months of data. This is the main reason why semi-coherent search methods have been developed. These methods perform coherent searches over shorter stretches of data, called segments, and then combine the results with incoherent techniques.
This search covers waveforms from the entire sky, with frequencies in a 250 Hz range from 1249.717 Hz to 1499.717 Hz, and with a first-order spin-down between −2.93 × 10 −9 Hz/s and 5.53 × 10 −10 Hz/s, similar to previous Einstein@Home searches. We use a "stack-slide" semi-coherent search procedure implemented with the GCT method [15,16]. In the left panel, we show the sky-grid points on the celestial sphere; the color-code traces the number of sky-grid points, N δ , as a function of equatorial latitude δ. The right panel is a polar plot of the northern equatorial hemisphere of the same sky-grid but with density scaled down by a factor of 4 to allow for better viewing. In the polar plot, θ = α and r = Cos(δ).
The data is divided into N seg segments, each spanning T coh in time. The coherent multi-detector F -statistic [11] is computed on each segment for all the points on a coarse λ c ≡ { f c , f c , α c , δ c } signal waveform parameter grid, and then results from the individual segments are summed, one per segment, to yield the final core detection-statistic F , as shown in (1); α, δ are the equatorial sky coordinates of the source position, while f and f are the frequency and first-order spin-down of the signal respectively. Depending on which λ c parameter points are taken on the coarse grid for each segment in this sum, the result will approximate the detectionstatistic computed on a λ f parameter point on a finer grid: In a "stack-slide" search in Gaussian noise, N seg × 2F follows a χ 2 4N seg chi-squared distribution with 4N seg degrees of freedom.
The most important search parameters are then: N seg , T coh , the signal parameter search grids λ c , λ f , the total spanned observation time T obs , and finally the ranking statistic used to rank parameter space cells i.e. 2F .
The grid-spacing in frequency δf and spin-down δ f are constant over the search range. The same frequency spacing and sky grid is used for the coherent analysis and in the incoherent summing. The spin-down spacing of the incoherent analysis is finer by a factor of γ with respect to that of the coherent analysis. In table 1, we summarize the search parameters. The sky-grid for the search is constructed by tiling the projected equatorial plane uniformly with squares of edge length d sky . The length of the edge of the squares is a function of the frequency f of the signal, and parameterized in terms of a so-called sky-mismatch parameter (m sky ) as where, τ E = 0.021 seconds and m sky = 0.3, also given in table 1. The sky-grids are constant over 10 Hz-wide frequency bands, and are calculated for the highest frequency in the band. In figure 1, we illustrate an example of the sky-grid. The total number of templates in 50 mHz bands as a function of frequency is shown in figure 2. This search explores a total of 5.6 × 10 16 waveform templates across the The search is divided into work-units (abbrev. WU), each searching a very small sub-set of template waveforms. The WU are sent to Einstein@Home volunteers and each WU occupies the volunteer/host computer for roughly 6 hours. One such WU covers a 50 mHz band, the entire spin-down range, and 139-140 points in the sky. 6.4 million different WU are necessary to cover the whole parameter space. Each WU returns a ranked list of the most significant 10 4 candidates found in the parameter space that it searched.

Identification of Undisturbed Bands
In table 3, we list the central frequencies and bandwidths of SFT data known to contain spectral lines from instrumental artefacts. These frequency regions were identified before the Einstein@Home run, and we were able to replace the corresponding data with Gaussian noise matching the noise level of neighbouring quiet bands. Consequently, some search results have contributions from this 'fake data'. The intervals in signal-frequency where the search results come entirely from fake data are indicated as All Fake Data in table 4. In these intervals of signal-frequency, we effectively do not have search results. The other three columns in table 4 provide signal-frequency intervals where results might have contributions from fake data. In these regions, depending on the signal parameters, the detection efficiency might be affected.
Despite the removal of known disturbances from the data, it still contains unknown noise artefacts producing 2F values that do not follow the expected distribution for Gaussian noise. These artifacts usually have narrowband characteristics; we identify such 'disturbed' signalfrequency intervals in the search results and exclude them from further post-processing analysis.

Quantity
Value  The benefit of such exclusions is that, in the remaining 'undisturbed' bands, we can rely on semi-analytic predictions for the significance of the observed 2F values, and we can set a uniform detection criterion across the entire parameter space. It is true that we forego the possibility of detecting a target signal in the 'disturbed' frequency intervals. However, to perform reliable analysis in these intervals, ad-hoc studies and tuning of the procedures would need to be set up. These additional procedures would require as much, if not longer, than the time spent on the 'undisturbed' data set. Moreover, since the 'undisturbed' intervals in data comprise over 95% of the total data, we believe this is a reasonable choice. In the future, a focused effort on the analysis of 'disturbed bands' could attempt to recover some sensitivity in those regions. The identification of undisturbed bands is carried out via a visual inspection method. This visual inspection of the data is performed by two scientists who look at various distributions of the 2F values in the { f , f } parameter space in 50 mHz bands. They rank these 50 mHz bands with 4 numbers: 0,1,2,3; a '0' Figure 3: We plot the color-coded 2F values on the z-axis in three 50 mHz bands. The top-most band is marked as "disturbed"; the middle band is an example of an "undisturbed" band; the bottom-most band is an example of an "undisturbed" band but containing a simulated continuous gravitational wave signal. ranking marks the band as "undisturbed", a '3' ranks the band as 'disturbed", and rankings of '1' or '2' mark the band as "marginally disturbed". A 50 mHz band is eventually considered to be undisturbed if it is marked as '0' by both scientists. The criteria used for this inspection are based on training-sets of real data containing simulated signals. These criteria are designed to exclude disturbed set of results while retaining data sets with signal-like properties, and to err on the side of being conservative in terms of not falsely dismissing signals.A significant part of this visual inspection work can be automated [20], but at the time of this search, the procedure had not been fully tested and tuned. In figure 3, we empirically illustrate these criteria using three examples. Following this procedure, 3% of the total 5000 50 mHz bands are marked as "disturbed" by visual inspection. These excluded bands are listed in table 5 (Type D), together with the 50 mHz bands excluded as a result of the cleaning of known disturbances above (Type C), i.e. marked as "All Fake Data" in table 4. In consequence to these exclusions, there exist 0.5 Hz bands comprising results from less than ten 50 mHz bands. We define 'filllevel' as the percentage of 50 mHz bands that contribute to the results in 0.5 Hz intervals, where 100% fill-level signifies contribution by all ten 50 mHz bands. In figure  7, we show the distribution of fill-levels for the 0.5 Hz bands searched.
In figure 4, we plot the loudest observed candidate i.e. the candidate with the highest 2F value in each 0.5 Hz band in the search frequency range. The loudest candidate in our search has a detection-statistic value of 2F = 5.846 at a frequency of roughly 1391.667 Hz. In order to determine the significance of this loudest candidate, we compare it to the expected value for the highest detection-statistic in our search. In order to determine this expected value, we have to estimate the number of independent trials performed in the search i.e. total number of independent realizations of our detection-statistic 2F .
The number of independent realizations of the detection-statistic, N trials , in a search through a bank of signal templates is smaller than the total number of searched templates, N templates . We estimate N trials as a function of frequency in 10 Hz frequency intervals. In each of these 10 Hz intervals, we fit the distribution of loudest candidates from 50 mHz bands to the expected distribution [1], and obtain the best-fitted value of N trials . We perform this calculation in 10 Hz intervals since the sky-grids, along with N templates , are constant over 10 Hz frequency intervals. In figure 5, we plot the ratio R = N trials /N templates , as a function of frequency.
With R( f ) in hand, we evaluate the expected value for the loudest detection-statistic (2F exp ) in 0.5 Hz bands, and the standard deviation (σ exp ) of the associated distribution using (5)-(6) of [1], with N seg = 205 and N trials = R N templates . Based on these values, we can estimate the significance of the observed loudest candidates (denoted by 2F Loud ) as the 'Critical Ratio' (CR), (3) Figure 5: Plotted ratio R = N trials /N templates as a function of frequency in 10 Hz intervals. The error bars represent the 1-σ statistical errors from the fitting procedure described in the text. In figure 6, we plot the CR values of the observed loudest candidates in 0.5 Hz bands as a function of frequency (top panel) and their distribution (bottom panel).
In this search, the overall loudest candidate with 2F = 5.846 is also the most significant candidate, with CR = 3.05. A deviation of 3.05σ from the expected 2F value would not be significant enough to claim a detection if we had only searched a single 0.5 Hz band; in fact, it is even less significant considering the fact that a total of 485 0.5 Hz bands were searched. We define the p-value associated with a CR as the probability of observing that particular value of CR or higher by random chance in a search over one 0.5 Hz band, performed over N trials independent trials using N seg segments. In figure 8, we see that the distribution of p-values associated with the loudest observed candidates in 0.5 Hz bands is consistent with what we expect from the noise-only scenario across the explored parameter space. In particular, we see in figure 8 that across 485 0.5 Hz bands searched by our set up, we expect 2.3 ± 1.5 candidates at least as significant as CR = 3.05 (p-value bin 10 −2 for that band) by random chance, which makes our observed loudest candidate completely consistent with expectations from the noise-only case.

Upper-limits
Our search results do not deviate from the expectations from noise-only data. Hence, we set frequentist upperlimits on the maximum gravitational wave amplitude, h 90% 0 , from the target source population consistent with this null result at 90%-confidence in 0.5 Hz bands. Here, h 90% 0 is the gravitational wave amplitude for which 90% of the target population of signals would have produced a value of the detection statistic higher than the observed value.
Hence, we set frequentist upper-limits on the maximum detectable gravitational wave amplitude, h 90% 0 , at 90%-confidence in 0.5 Hz bands. Here, h 90% 0 is the gravitational wave amplitude for which 90% of a population of signals with parameter values in our search range would have been produced a value of the detection statistic higher than the observed one in that search range.
Ideally, in order to estimate the h 90% 0 values in each 0.5 Hz band across the 250 Hz signal-frequency search range, we would perform Monte-Carlo injection-andrecovery simulations in each of those bands. However, this is computationally very intensive. Therefore, we perform Monte-Carlo simulations in six 0.5 Hz bands spread evenly across the 250 Hz-wide frequency range, and in each of these six bands labeled by the index k, we estimate the h 90%,k 0,CR i upperlimit value corresponding to eight different CR i 'significance bins' for the putative observed loudest candidate: (0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5). In each of these six bands and for each of the eight detection criteria, we calculate the so-called 'sensitivity-depth', defined in [1]: . Lastly, we average these sensitivity-depths over the six bands and derive the average sensitivity-depth D 90% where, CR i (l) is the 'significance bin' i corresponding to the loudest observed candidate in the l-th frequency band, and S h ( f l ) is the average amplitude spectral density of the data in that band, measured in Hz −1/2 . The uncertainties on the h 90% 0 upper-limit values introduced by this procedure amount to roughly 10% of the nominal h 90% 0 upper-limit value. The final h 90% 0 upper-limit values for this search, including an additional 10% calibration uncertainty, are given in table 2, and shown in figure 9.
Note that we do not set upper limits in 0.5 Hz bands where the results are entirely produced with fake Gaussian data inserted by the cleaning procedure described in section 4; h 90% 0 upper-limit values for such bands do not appear either in table 2, or in figure 9.
Moreover, there also exist 50 mHz bands that contain results contributed by entirely fake data as a result of the cleaning procedure, or that have been excluded from the analysis because they are marked as 'disturbed' by the visual inspection method described in section 4. We mark the 0.5 Hz bands which host these particular 50 mHz bands with empty circles in figure 9. In table 5, we provide a complete list of such 50 mHz bands, highlighting that the upper-limit values do not apply to these bands. Finally, we note that, because of the cleaning procedure, there exist signal-frequency bands where the search results may have contributions from some fake data. We list these signal-frequency ranges in table 4. In line with the remarks in section 4, and for the sake of completeness, table 4 also contains the cleaned bands that featured under Type C in table 5, under the column header "All Fake Data".

Conclusions
This search did not yield any evidence of continuous gravitational waves in the LIGO 5th Science Run data in the high-frequency range of 1250-1500 Hz. The lowest value for the upper-limit is 5.0×10 −24 for signal frequencies between 1253.217-1255.217 Hz. We show in figure  9 that these h 90% 0 upper-limits are about 33% higher than the upper-limits1 set [10] in the same frequency range but using S6 data . In this frequency range, the S6 run data is about a factor 2.4 more sensitive compared to the S5 data used in this search.
We can express the h 90% 0 upper-limits as bounds on the maximum distance from Earth within which we can exclude a rotating compact object emitting continuous gravitational waves at a given frequency f due to a fixed and non-axisymmetric mass quadrupole moment, characterised by I, with I being the principal moment of inertia, and the ellipticity of the object. The 'GWspindown' is the fraction of spin-down, x| f |, responsible for continuous gravitational wave emission [14]. The ellipticity ( ) of the compact object necessary to sustain such emission is given by where, c is the speed of light, G is the Gravitational constant. Moreover, since the gravitational wave amplitude for an object at a distance d source , with an ellipticity given by (5), is expressed as We can recast the h 90% 1The upper-limit values of [10] have been re-scaled according to [19] in order to allow a direct comparison with our h 90% 0 upper-limit results. Figure 9: 90%-confidence upper-limits on the gravitational wave amplitude for signals with frequency within 0.5 Hz bands, over the entire sky, and within the spin-down range of the search described in section 3. The empty circular markers denote 0.5 Hz bands where the upper-limit value does not hold for all frequencies in that interval; the list of corresponding excluded frequencies is given in table 4. For reference, we also plot the upper-limit results (with non-circular markers) from the only other high-frequency search [10], on significantly more sensitive S6 data. It should be noted that the upper-limits from the PowerFlux search [10] are set at 95%-confidence rather than 90%-confidence level as in this search, but refer to 0.25 Hz bands rather than 0.5 Hz bands. 1.0 × 10 −9 Hz/s. This value is well below the maximum elastic deformation that a relativistic star could sustain, see [12] and references therein.
The search presented here is probably the last allsky search on S5 data, and by inspecting the higher frequency range for continuous gravitational wave emission, it concludes the Einstein@Home observing cam-paign on this data. Consistent with the recent results on S6 data [10], we also find no continuous GW signal in the S5 data. However, mechanisms for transient or intermittent GW emission have been proposed [18,17,13] which would not a priori exclude a signal that is "ON" during the S5 run and "OFF" during the S6 run. The estimates for the time-scales, frequencies, and spin-downs of continuous gravitational wave signals from isolated neutron stars lasting weeks to months span a very broad range of values -orders of magnitude. There are several different mechanisms that could sustain such emission at a level that this search could have detected, and with spin-down values consistent with the total energy emitted in the process, and with the spin-down range spanned by this search.

Acknowledgments
Maria Alessandra Papa, Bruce Allen and Xavier Siemens gratefully acknowledges the support from NSF PHY Grant 1104902. All the post-processing computational work for this search was carried out on the ATLAS super-computing cluster at the Max-Planck-Institut für Gravitationsphysik/ Leibniz Universität Hannover. We also acknowledge the Continuous Wave Group of the LIGO Scientific Collaboration for useful discussions, and in particular, its chair Keith Riles for his careful reading of the manuscript. This paper has been assigned the LIGO Document number P1600196. * * * Appendix: Tabular Table 4: Signal-frequency ranges where the results might have contributions from fake data. When the results are entirely due to artificial data, the band is listed in the "All Fake Data" column; bands where the results comprise of contributions from both fake and real data are listed in the other three columns. The "Mixed (Left)" and "Mixed (Right)" columns are populated only when there is a matching "All Fake Data" entry, which highlights the same physical cause for the fake data, i.e. the cleaning.
The "Mixed (Isolated)" column lists isolated ranges of mixed data. The list of input data frequencies where the data was substituted with artificial noise are given in table 3.  Table 5: 50 mHz search-frequency bands that were identified as "disturbed" based on Visual Inspection (Type D), or where the results were produced from "All Fake Data", as detailed in table 4 (Type C). Both sets of bands (Type D and C) were excluded from the analysis. The first two columns list the starting frequency of the first and last 50 mHz band in the contiguous range of excluded bands.